Minimization of torque ripple

ABSTRACT

Systems and methods of operating an electric motor including a first linear actuator including a first coil, a second linear actuator including a second coil, a rotational shaft, a cam assembly mounted on said rotational shaft for translating linear movement of the first and second linear actuators to rotational movement of the rotational shaft. The first coil is driven with a first signal that produces a first radially-directed force. The second coil is simultaneously driven with a second signal that produces a second radially-directed force, wherein the first radially-directed force is represented by a sine function and the second radially-directed force is represented by the sine function phase shifted by ±π/2 radians.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a divisional of prior U.S. patent application Ser.No. 13/587,467, filed Aug. 16, 2012 which claims benefit or priority toU.S. Provisional Application No. 61/524,089, filed Aug. 16, 2011, theentire disclosures of which are hereby incorporated by reference intheir entirety.

TECHNICAL FIELD

This invention relates generally to the control of electric motors andmore specifically to the control of linear Lorentz-Type actuator motors.

BACKGROUND OF THE INVENTION

Lorentz-type motors exploit the basic principle that a charged particlemoving in a magnetic field experiences a force in a directionperpendicular to the direction of movement. Stated mathematically:F=qvXB, where F is force, q is the charge of the charged particle, v isthe instantaneous velocity of the particle, and B is the magnetic field.So, if a current is flowing through a wire and a magnetic field isapplied in perpendicular direction, the wire experiences a force tryingto move it sideways.

A simple configuration that harnesses this principle is a coilencircling a magnetic core made of permanent magnets. The coil, referredto as the actuator, is arranged to be capable of sliding back and forthalong the length of the magnetic core or magnetic stator. In thatconfiguration, flowing a current though the coil results in a force onthe coil pushing it in one direction along the length of the magneticcore. Reversing the direction of current flow causes the coil to move inthe opposite direction. The magnitude of the current determines thestrength of the force. And the shape of the current waveform determineshow the force changes over time. With such an arrangement, by applyingan appropriate current waveform to the coil, one can make the coil moveback and forth along the magnetic core in a controlled manner. Thecontrolled movement of the actuator can, in turn, be used to performwork.

BRIEF DESCRIPTION OF THE DRAWINGS

FIGS. 1A-1B illustrate a rotary motor in a wheel.

FIG. 1C illustrates a magnetic stator assembly.

FIGS. 2A-2C illustrate components of the rotary motor of FIG. 1 invarious stages of motion.

FIG. 2D illustrates an exemplary shape of a cam.

FIG. 3 is a cam profile based on an Archimedes Spiral

FIG. 4 is a plot of cam radial position as a function of angle.

FIG. 5 is a sinusoidal cam profile.

FIG. 6 is a plot of cam radial position as a function of angle for thecam shown in FIG. 5.

FIGS. 7A-7B illustrate an arrangement of two hub motors for one wheel.

FIGS. 8A-8B illustrate views of two rotationally offset cams.

FIGS. 9A-9B illustrate a disc of an example rotary device coupled to arim of a wheel.

FIG. 10 illustrate two cam profiles in quadrature.

FIG. 11 illustrate the symmetry features of a torque function.

FIG. 12 illustrates a piecewise quadratic torque function.

FIG. 13 illustrates a piecewise quadratic profile for the firstderivative of the cam profile.

FIG. 14 illustrates a non-trigonometric force function compared to atrigonometric force function.

FIG. 15 illustrates a non-trigonometric cam profile compared to atrigonometric cam profile.

FIG. 16 is an exemplary control system for providing constant torque.

FIG. 17 shows a load cell for directly measuring the Lorentz force.

FIG. 18 is an exemplary control system which employs feedback for forceprofile generation.

DETAILED DESCRIPTION

The subject of this application is the design and operation of ahub-mounted motor assembly so as to minimize torque ripple. Thehub-mounted motor is a linear Lorentz-type actuator motor. Beforediscussing the design and operation of the hub-mounted motor assembly, abrief review of the linear Lorentz-type actuator motor will bepresented. A more detailed discussion can be found in U.S. Ser. No.12/590,495, entitled “Electric Motor,” and incorporated herein in itsentirety by reference.

The Linear Lorentz-Type Actuator Motor

The linear Lorentz-type actuator motor is a rotary device 100 that ismounted inside a wheel on a vehicle, as illustrated in FIG. 1A. Rotarydevice 100 includes a magnetic stator assembly 120, opposedelectromagnetic actuators 110 a, 110 b, and a linear-to-rotary converter(e.g., oval-shaped cam) 105. Rotary device 100 is attached to thechassis of a vehicle, for example, at a point on the far side of thewheel (not shown). Rotary device 100 is attached to the wheel via cam105 using a circular support plate, for example, which has been removedto show the inside of the wheel. Such a plate is attached to both therim of the wheel and cam 105 using fasteners, such as bolts. The wheeland cam support plate rotate relative to a hub 145 about a bearing 150.

FIG. 1B shows rotary device 100 from the side of the wheel 140 with thetire and some other components removed. The core of rotary device 100includes cam 105, two opposed electromagnetic actuators 110 a, 110 b,and a magnetic stator assembly 120. Electromagnetic actuators 110 a, 110b each house a coil 115 a, 115 b that encircles magnetic stator assembly120. Magnetic actuators 110 a, 110 b is arranged to reciprocate relativeto magnetic stator assembly 120 when an appropriate drive signal isapplied to coils 115 a, 115 b. One electromagnetic actuator 110 a isshown having a housing 155 a surrounding its coil 115 a and the otherelectromagnetic actuator 110 b is shown with its housing removed to showits coil 115 b.

Magnetic stator assembly 120 depicted in FIG. 1B is oriented verticallyand includes a plurality of magnetic stators 125 a, 125 b, each of whichincludes multiple individual permanent magnets oriented so that theirmagnetic moments are perpendicular to the axis of magnetic statorassembly 120. When current is applied to coils 115 a, 115 b of theelectromagnetic actuators 110 a, 110 b (e.g., alternating current),actuators 110 a, 110 b are forced to move vertically along magneticstator assembly 120 due to the resulting electromagnetic forces (i.e.,the Lorentz forces). As is well known, when a coil carrying anelectrical current is placed in a magnetic field, each of the movingcharges of that current experiences what is known as the Lorentz force,and collectively they create a net force on the coil. The direction ofmovement and force generated is controlled by the polarity and amplitudeof the current induced in the coil.

Rotary device 100 also includes a plurality of shafts 130 a, 130 b,coupled to a bearing support structure 165. Electromagnetic actuators110 a, 110 b slide along the shafts using, for example, linear bearings.Attached to each electromagnetic actuator 110 a, 110 b is a pair offollowers 135 a-d that interface with cam 105 to convert their linearmotion to rotary motion of the cam. To reduce friction, followers 135a-d freely rotate so as to roll over the surfaces of cam 105 during theoperating cycle. Followers 135 a-d are attached to electromagneticactuators 110 a, 110 b via, for example, the actuators' housings. Aselectromagnetic actuators 110 a, 110 b reciprocate, the force exerted byfollowers 135 a-d on cam 105 drives cam 105 in rotary motion.

FIG. 1C illustrates magnetic stator assembly 120 with two magneticstators 125 a, 125 b. Magnetic stators 125 a, 125 b each includemultiple magnets. For example, magnetic stator 125 a includes, on oneend surface portion, eight magnets 160 a-h. All of the magnets 160 havetheir magnetic moments oriented perpendicular to the surface on whichthey are mounted and in the same direction.

FIGS. 2A-C illustrate components of rotary device 100 in action,including the rotary device's electromagnetic actuators 110 a, 110 b(with associated coils 115 a, 115 b and followers 135 a-d) and cam 105moving relative to the magnetic stator assembly 120 (includingassociated magnetic stators 125 a, 125 b). The rim, the wheel, and thehousings by which the followers are attached to the coils are not shownin these figures. As illustrated by FIGS. 2A-C, the reciprocal movementof the coils 115 a, 115 b in opposition drives cam 105 to rotate, which,in turn, causes a wheel attached to cam 105 to rotate. Coils 115 a, 115b are shown in FIG. 2A as being at almost their furthest distance apart.FIG. 2B shows that as coils 115 a, 115 b move closer to each other,coils 115 a, 115 b drive cam 105 to rotate in a clockwise direction,thereby causing any attached wheel to also rotate clockwise. In theexample device, the force exerted on cam 105 is caused by the outerfollowers 135 a, 135 c squeezing-in on cam 105. FIG. 2C shows that coils115 a, 115 b are even closer together causing further clockwise movementof cam 105.

After coils 115 a, 115 b have reached their closest distance to eachother and cam 105, in this case, has rotated ninety degrees, coils 115a, 115 b begin to move away from each other and drive cam 105 tocontinue to rotate clockwise. As coils 115 a, 115 b move away from eachother, inner followers 135 b, 135 d exert force on cam 105 by pushingoutward on cam 105.

It is noted that cam 105 is shown in the figures as an oval shape, butit may have a more complex shape, such as, for example, a shape havingan even number of lobes, as illustrated in FIG. 2D. The sides of eachlobe may be shaped in the form of a sine wave, a portion of anArchimedes spiral, or some other curve, for example. The number of lobesdetermines how many cycles the coils must complete to cause the cam torotate full circle. A cam with two lobes will rotate full circle upontwo coil cycles. A cam with four lobes will rotate full circle upon fourcoil cycles. Additionally, more lobes in a cam results in a highertorque.

Analysis of Torque

The motor consists of circular disk with an outer cam and inner cam. Twocam followers linked to a coil can create a radial force on the cam. Theforce exerted by the cam followers in turn creates a torque on the disk.

The idealized equation for the Torque T_(c)(θ) generated by the camfollower is given by the following equation:

$\begin{matrix}{{T_{c}(\theta)} = {{F_{c}(\theta)} \cdot \frac{{R_{c}(\theta)}}{\theta}}} & {{Eq}.\mspace{14mu} 1}\end{matrix}$

where F_(c)(θ) is the radial force generated by the cam follower, andR_(c)(θ) is the distance of the cam follower from the center of thedisk. As noted above, in the motor, the force is generated from acurrent running in a coil and interacting with a magnetic field.

If the force is constant throughout a half stroke and the slope definingthe position of the cam follower as a function of the wheel angle isalso constant, that produces a torque that is constant throughout thecycle. The two dimensional shape of the cam would then be as depicted inFIG. 3.

In FIG. (1), θ is the position (rotation) of the wheel in radians andthe disk has four lobes. The cam follower exerts a vertical force asindicated by the arrow. The position of the cam in polar coordinates isgiven by the curve shown in FIG. 4.

Although this cam profile easily lends itself to a drive signal thatyields a constant torque, it presents two major drawbacks: the need toinstantaneously change the coil velocity at the end of the cam motionand the need to instantaneously change the current that generates theforce exerted by the cam.

The approximate equation giving the force required to accelerate anddecelerate the coil is:

$\begin{matrix}{F_{r} = {M_{c} \cdot \left\lbrack {{\frac{^{2}}{\theta^{2}}{{R_{c}(\theta)} \cdot \left( {\frac{}{t}\theta} \right)^{2}}} + {\frac{}{\theta}{{R_{c}(\theta)} \cdot \frac{^{2}}{t^{2}}}\theta}} \right\rbrack}} & {{Eq}.\mspace{14mu} 2}\end{matrix}$

where M_(c) is the mass of the coil. At the extremes of the strokemotion, the term

$\frac{^{2}{R_{c}(\theta)}}{\theta^{2}}$

is theoretically infinite, which means in practice that the coil wouldundergo an unacceptably high shock due to the abrupt deceleration andacceleration.

Instantaneously changing the current also presents technical challenges,given that the current flows through a coil with a significantinductance. A linear approximation of the voltage required to change thecurrent in the coil is given by:

$\begin{matrix}{{V_{c}(t)} = {{R_{c} \cdot {I_{c}(t)}} + {{L_{c} \cdot \frac{}{t}}{I(t)}} + {V_{emf}(t)}}} & {{Eq}.\mspace{14mu} 3}\end{matrix}$

where V_(c)(t) is the voltage required across the coil as a function ofthe required change in coil current I_(c)(t), R_(c) is the coilresistance, L_(c) is the coil inductance and V_(emf)(t) is the backelectromotive force generated by the coil as it moves through a changingmagnetic field. Here again, given the discontinuity in the current thevoltage across the coil would tend to infinity.

One partial solution is to change the cam profile such that its secondderivative is finite and continuous at all points, i.e., it has a thirdorder derivative. One example of such a cam profile would be asinusoidal shape. In such a case, the cam profile would be given by theequation:

R _(c)(θ)=R ₀ +A _(c)·sin(n ₁·θ)  Eq. 4

where R₀ is the circle around which the cam evolves (mean position),A_(c) is the cam amplitude and n₁ is the number of lobes or number ofstrokes per revolution. The cam profile then looks like what is shown inFIG. 5. And in polar coordinates, the cam profile is shown in FIG. 6.

In this case the force to be generated by the coil is given by:

$\begin{matrix}{{F_{c}(\theta)} = \frac{T_{c}(\theta)}{\frac{}{\theta}{R_{c}(\theta)}}} & {{Eq}.\mspace{14mu} 5}\end{matrix}$

However, the derivative of the cam position R_(c)(θ) is null at the endof the strokes, hence the required force would also diverge to infinity.This remains true for any cam profile. If there is only one cam, thecorollary is that it would not be self-starting if the initial positionoccurs when the cam follower is at the end of a stroke.

Using multiple cams such that their null points are spaced apartcircumvents the problem. In the simplest example, there would be twocams on a disk, each on opposite side.

An Exemplary Embodiment

A wheel which implements this approach is shown in FIGS. 7A-B. In thiscase the tire of the wheel, and some other components, have been removedfor clarity. There are two rotary devices 1500 and 1600, one mounted oneach side of a central disc 1635. The rotary devices are similar to therotary devices described above. Rotary device 1500 includes a pair ofelectromagnetic actuators 1510 a, 1510 b, and a magnetic stator assembly1520. Similarly, rotary device 1600 includes a pair of electromagneticactuators 1610 a, 1610 b, and a magnetic stator assembly 1620. Each ofmagnetic stator assemblies 1520, 1620 includes two magnetic stators 1515a, 1515 b, 1615 a, 1615 b, which include magnetic flux return paths 1640a-d and magnets (e.g., 1630 a, 1630 b). The housings surrounding thecoils of the electromagnetic actuators 1510 a, 1510 b, 1610 a, 1610 bare not shown. Each coil reciprocates along four arrays of magnets,which, as described above, may include multiple magnets. Two of themagnet arrays are located inside the coil (e.g., inner magnetic statorcomponent 1630 b) and two are located outside the coil (e.g., outermagnetic stator component 1630 a). Each set of magnets are mounted to amagnetic flux return path 1640 a-d.

The disc 1635 includes two cams, one on either side of the disc 1635. Inthis example, each cam of the device is in the form of a grove thatincludes an inner surface 1605 a and an outer surface 1605 b. Coupled toelectromagnetic actuators 1510 a and 1510 b are two pairs of followers1625 a, 1625 b, the different followers of each pair interfacing with arespective surface 1605 a, 1605 b of the cam. Electromagnetic actuators1610 a and 1610 b are similarly coupled to followers. As the coils movetowards each other, one of the followers of each electromagneticactuator 1510 a, 1510 b exerts force on the inner surface 1605 a of thecam. As the coils move away from each other, the other follower exertsforce on the outer surface 1605 b of the cam.

FIG. 7B illustrates a different view of the rotary devices. It should beapparent that each pair of electromagnetic actuators (pair 1510 a, 1510b and pair 1610 a, 1610 b) are at different phases of reciprocation.This is because, in the example device, the cams on either side of thedisc 1635 are rotationally offset from each other by, for example,forty-five degrees. This helps to prevent the actuators from stopping ata point on the cams from which it would be difficult to again start.Thus, if one pair of actuators stops on a “dead-spot” of its respectivecam, the other pair of actuators would not be at a dead-spot. FIG. 7Balso illustrates an arrangement of the coils and magnetic statorcomponents. For example, magnetic stators components 1630 b and 1630 care located inside the coil of actuator 1610 a, and magnetic statorscomponents 1630 a and 1630 d are located outside the coil.

FIG. 8A illustrates two rotationally offset cams 1505, 1606. The cams1505, 1606 are part of or are mounted on a disc 1635. One cam 1505 is onone side of the disc 1635, and the other cam 1606 is on the oppositeside, as indicated by the dashed line. In some devices the cam may beoffset by forty-five degrees, for example. The cams 1505, 1606 have aneven number of lobes, e.g. 2, 4, 6 etc.. Cams having two lobes areoffset by 45 degrees. Cams having four lobes are offset by 22.5 degrees.

FIG. 8B illustrates a vertical cross-section of a disc 1635 with tworotationally offset cams, each having in inner surface 1605 a, 1605 cand an outer surface 1605 b, 1605 d. Due to the offset, the innersurfaces 1605 a, 1605 c are not in line with each other. Likewise, theouter surfaces 1605 b, 1605 d are also not in line with each other.

FIG. 9A illustrates how the disc 1635 of the example rotary device iscoupled to the rim 1705 of a wheel. The rim 1705 consists of one pieceto which the disc 1635 is affixed using fasteners, such as bolts, alongan inner ring 1715. Alternatively, the rim 1705 may include two parts1710 a, 1710 b that bolt together along ring 1715. When fastenedtogether, the two parts 1710 a, 1710 b form a full rim 1705 with innerring 1715. A tire is then be mounted to the rim 1705. FIG. 9B shows howthe disc 1635 is fastened to the inner ring 1715 of the disc 1635.

Minimizing Torque Ripple

Returning to the description of the technique for minimizing torqueripple, we direct the reader's attention to an example using two fourlobe cams, which is illustrated in FIG. 10.

Note that the profiles are not mirror images but are in quadrature andthat they consist of the same basic profile R_(c)(x) but “shifted” withrespect to each other. Assuming that n_(c) is the number of lobes in thecam, i.e. the number of times that a basic function R_(c)(x) is repeatedwithin one full circle, and that the derivative of the cam profile isgiven by Ψ_(c)(n_(c)·θ), the equation for the torque provided by thesummation of the torques Φ(θ) of the individual cams would be:

T=Φ(n _(c)·θ)+Φ(n _(c)θ+Δ)  Eq. 6

T=F _(c)(n _(c)·θ)·Ψ(n _(c)·θ)+F _(c)(n _(c)·θ+Δ)·Ψ(n _(c)·θ+Δ)  Eq. 7

And typically Δ corresponds to one quarter of a lobe, i.e.

$\Delta = {\frac{\pi}{2}.}$

This leads to the following question: what is the family of functionsΦ(θ)=F_(c)(θ)·Ψ_(c)(θ) such that total torque is constant (i.e., ripplefree) when at least two out-of-phase cams and actuators are used. Thisequation implies that the function Φ(θ) has a periodicity of 2·Δ:

Φ(θ)=T−Φ(θ+Δ)  Eq. 8

Φ(θ+Δ)=T−Φ(θ+2·Δ)  Eq. 9

Substituting:

Φ(θ)=T−(T−Φ(θ+2·Δ))=Φ(θ+2·Δ)  Eq. 10

Which is as required, i.e. the function Φ(θ) is periodic, with a periodthat is double that of the period of one full cam cycle. This stillleaves the ensemble of functions quite large. For reasons of symmetry,it is reasonable to require that:

Φ(θ)=Φ(−θ)  Eq. 11

And we can also assume that the cam reaches its extremum at θ=0 and θ=Δ.Hence, the class of functions Φ(θ) that we are seeking has the followingproperties:

-   -   1. Φ(θ)=0    -   2. Φ(Δ)=Tmax where Tmax is the maximum torque    -   3. Φ(θ) is symmetrical with respect to the θ=0 vertical axis    -   4. Φ(θ) is symmetrical with respect to the point (Δ/2, Tmax/2).    -   5. Φ(θ) is continuous.        The general shape of this function is given in FIG. 11.

We also know that the function Φ(θ) is the product of two otherfunctions, F_(c)(θ) and Ψ(θ), where Ψ(θ) must have a first orderderivative, such that the cam profile given by:

R _(c)(θ)=∫Ψ(θ)dθ  Eq. 12

Intuitively it would also be desirable that the functions F_(c)(θ),R_(c)(θ) and Ψ(θ) have the same symmetry. Therefore one reasonablequestion to ask is would there be a function such that F_(c)(θ) and Ψ(θ)are the same function? In this case:

T=Ψ ²(n _(c)·θ)+Ψ²(n _(c)·θ+Δ)  Eq. 13

which is the basic equation of a right triangle.

Here the components in quadrature can be viewed as the sides of a righttriangle, such that one is a sine of an angle and the other one thecosine of the angle:

$\begin{matrix}{\mspace{79mu} {{\cos^{2}(\phi)} = {\Psi^{2}\left( {n_{c} \cdot \theta} \right)}}} & {{Eq}.\mspace{14mu} 14} \\{{\cos \left( {\phi + \Delta} \right)} = {{\cos \left( {\phi + \frac{\pi}{2}} \right)} = {{{{\cos (\phi)}{\cos \left( \frac{\pi}{2} \right)}} - {{\sin (\phi)}{\sin \left( \frac{\pi}{2} \right)}}} = {- {\sin (\phi)}}}}} & {{Eq}.\mspace{14mu} 15} \\{\mspace{79mu} {{\therefore{\cos^{2}\left( {\phi + \Delta} \right)}} = {\sin^{2}(\phi)}}} & {{Eq}.\mspace{14mu} 16}\end{matrix}$

In conclusion, if the shape of the cam is a sine function, itsderivative is a cosine function, its derivative is a sine function, andif the current waveform is also a sine function then the two componentsin quadrature sum up to a constant torque with no ripple.

In principle, there is an infinite number of functions F_(c)(θ),R_(c)(θ) and Ψ(θ) leading to a constant torque. In practice, the choiceis rather limited, given that we must have:

$\begin{matrix}{{F_{c}(\theta)} = \frac{\Phi (\theta)}{\Psi (\theta)}} & {{Eq}.\mspace{14mu} 17}\end{matrix}$

And when both Φ(θ) and Ψ(θ) tend to zero, the ratio must also convergeto zero. We also require to a first order derivative. It becomes anon-trivial exercise to find other functions besides the sinusoidal typefunction to meet these criteria, and they typically end up very close toa trigonometric function. However, in the modern day of microprocessorbased digital control where computation time is a prime considerationsuch alternate functions might have a benefit.

One of the simplest examples of an alternate approach would be to usepiece-wise quadratic functions for Φ₁(θ) and Ψ_(i)(θ), as given in thisMathCAD recursive representation, omitting for the time being the numberof lobes in the equations:

$\begin{matrix}\left. \begin{matrix}{{\Phi_{1}(\theta)}:=} \\\; \\\; \\\; \\\; \\\; \\\; \\\;\end{matrix} \middle| \begin{matrix}\left. z\leftarrow{\theta } \right. \\\left. z\leftarrow{{mod}\left( {z,\pi} \right)} \right. \\\left. y\leftarrow{{{\frac{8}{\pi^{2}} \cdot z^{2}}\mspace{14mu} {if}\mspace{14mu} z} \leq \frac{\pi}{4}} \right. \\{otherwise} \\\left| \begin{matrix}\left. y\leftarrow{{1 - {{\Phi_{1}\left( {\frac{\pi}{2} - z} \right)}\mspace{14mu} {if}\mspace{14mu} z}} \leq \frac{\pi}{2}} \right. \\\left. y\leftarrow{{\Phi_{1}\left( {\pi - z} \right)}\mspace{14mu} {otherwise}} \right.\end{matrix} \right. \\\left. y\leftarrow y \right.\end{matrix} \right. & {{Eq}.\mspace{14mu} (18)}\end{matrix}$

where θ₁(θ) is shown in FIG. 12.

$\begin{matrix}\left. \begin{matrix}{{\Psi_{1}(x)}:=} \\\; \\\; \\\; \\\; \\\; \\\; \\\;\end{matrix} \middle| \begin{matrix}\left. z\leftarrow{x} \right. \\\left. z\leftarrow{{mod}\left( {z,{2\pi}} \right)} \right. \\\left. y\leftarrow{{1 - {{\frac{4}{\pi^{2}} \cdot \left( {\frac{\pi}{2} - z} \right)^{2}}\mspace{14mu} {if}\mspace{14mu} z}} \leq \frac{\pi}{2}} \right. \\{otherwise} \\\left| \begin{matrix}\left. y\leftarrow{{{\Psi_{1}\left( {\pi - z} \right)}\mspace{14mu} {if}\mspace{14mu} z} \leq \pi} \right. \\\left. y\leftarrow{{- {\Psi_{1}\left( {z - \pi} \right)}}\mspace{14mu} {otherwise}} \right.\end{matrix} \right. \\\left. y\leftarrow{y \cdot {{sign}(x)}} \right.\end{matrix} \right. & {{Eq}.\mspace{14mu} (19)}\end{matrix}$

where Ψ₁(θ) is shown in FIG. 13.

The resulting force profile F₁ calculated from the ratio of Φ₁ to Ψ₁ isoutlined in FIG. 14, and compared to a trigonometric function.

Finally, the CAM profile R₁ is computed from the integral of Ψ₁ andcompared to a trigonometric function in FIG. 15.

Although difficult to prove, it is to be expected that all CAM shapesand force profiles that are well behaved in terms of symmetry andsmoothness would all be very close in shape to trigonometric functions.Only two CAMs in quadrature were analyzed here, the same approach couldbe used for other even numbers of CAMS.

In actual practice, although it is easy to generate a CAM with a precisetriangular function, it is more difficult to generate a force profilethat is a sinusoidal. For an idealized Lorentz force actuator assuming aconstant magnetic induction B, this would translate in generating anexact current profile with a sinusoidal function. However, in practicethe magnetic induction B is not constant and depends on the geometry ofthe permanent magnets used to generate the field. Furthermore, the fieldgenerated by magnets depends on the temperature and is also influencedby the current flowing in the motor coil. All of these effects must becarefully modeled to generate a current that truly minimizes ripple.

A typical control system is depicted in FIG. 16. It includes switchingpower electronics 500 which supplies a pulse width modulated drivesignal to the coils in the motor to produce the desired torque andspeed. The operation of switching power electronics 500 is controlledbased on models of the motor including a model 502 of the magneticinduction of the motor (i.e., the magnetic field seen by the coil as afunction of the position of the coil, the current in the coil, and thetemperature of the coil) and a model 504 which enables one to determinedthe voltage of a pulse width modulated signal that is necessary toproduce the desired drive current in the coils. Input for the modelscomes from a rotary encoder 506 which indicates the angular position ofthe cam or wheel, a conversion module 508 that converts the angularposition into a position of the coil or cam follower, and varioussensors in the motor supplying information about the motor's operatingconditions. Note that the model changes depending on operatingconditions and some the model needs to take these into account. Thevarious sensors include a motor temperature sensor 510, a current sensor512, a battery voltage sensor 514, and a coil voltage sensor 516.

From a wheel rotary encoder 506, the angular and radial position of thecoil and cam follower are calculated. From the cam follower position,the desired force to be generated by the coil is calculated from afunction F_(c)(θ). The desired current required to produce this force isequal to the current in the coil times the magnetic induction. Since themagnetic induction B is not exactly uniform, it has to be estimated frommodel 502 using the motor temperature, coil current, and relativeposition of the coil with respect to the permanent magnets.

The desired current is converted to a pulse modulation width. This isdone in two steps. First from the model of the coil dynamics, a voltagerequired across the coil to obtain the desired current in the coil iscalculated. Then, a model of the power electronics is required tocalculate the switching duty cycle based on the desired voltage, thesupply voltage, the actual current in the coil and the voltage acrossthe coil.

So far the control is all feed forward model based. However, the modelshave a certain level of inaccuracy, so feedback is used to correctbetween the desired current and measured current.

An alternative approach to generating the desired force profile is bymeasuring the force that is generated and directly controlling thatforce using a feedback control on current, as summarized in FIGS. 17 and18.

FIG. 17 illustrates how the Lorentz force generated by the coil can bedirectly measured. A load cell 700 is inserted between the coil 702 andthe cam follower 704. The cam follower itself can be subject to largeoff-axis forces from the reaction with the cam. However, forcetransducers can be designed to be largely insensitive to such lateralforces. Therefore, an accurate measurement of the axial force can beobtained from the load cell and then used in various control algorithmsto adjust the current such that the required force profile is generated.

FIG. 18 illustrates one of the many alternative control strategies thatcan be used to obtain the required force profile. The algorithmsdescribed above to estimate the current needed to get the desired forcecould be used in a feedforward manner. Then, the error between thedesired force and measured coil force would be feed to some otherfeedback control system which uses that measurement to make currentcorrections in order to obtain the desired force profile. The advantageof the feedback control approach is that it will tend to be more stablethan a purely feed forward approach.

Other embodiments are within the following claims.

1. A method of operating an electric motor including a first linearactuator including a first coil, a second linear actuator including asecond coil, a rotational shaft, a cam assembly mounted on saidrotational shaft for translating linear movement of the first and secondlinear actuators to rotational movement of the rotational shaft, saidmethod comprising: driving the first coil with a first drive signal thatproduces a first torque on the rotational shaft; simultaneously drivingthe second coil with a second drive signal that produces a second torqueon the rotational shaft, wherein the sum of the first torque and thesecond torque produces a total torque, and wherein the first drivesignal and the second drive signal are selected to result in the totaltorque being substantially ripple free throughout a complete rotation ofthe rotational shaft.
 2. The method of claim 1, wherein the first drivesignal varies periodically over a complete rotation of the rotationalshaft and the second drive signal varies periodically over the completerotation of the rotational shaft.
 3. The method of claim 2, wherein thefirst drive signal and the second drive signal each have a period of ncycles per a complete rotation of the rotational shaft, wherein n is aneven integer.
 4. The method of claim 3, wherein n equals
 4. 5. Themethod of claim 1, wherein the first drive signal is shifted in phaserelative to the second drive signal by ± $\frac{\pi}{2n}$ radians,wherein n is an integer.
 6. The method of claim 1, wherein the camassembly presents to the first linear actuator a first cam surfacehaving a first profile and presents to the second linear actuator asecond cam surface having a second profile, wherein a first derivativeof the first profile is represented by Ψ(n_(c)·θ) and a first derivativeof the second profile is represented by Ψ(n_(c)·θ+Δ), wherein θ is anangle of rotation of the rotational shaft, n, is an even integerrepresenting a number of cycles of the first profile the and secondprofile over a complete rotation of the rotational shaft, and Δ is aphase shift between the first profile and the second profile, whereinthe first drive signal causes the first linear actuator to generate aforce represented by F_(c)(n_(c)·θ) and the second drive signal causesthe second linear actuator to generate a force represented byF_(c)(n_(c)·θ+Δ), and wherein the first drive signal and the seconddrive signal are selected to causeF_(c)(n_(c)·θ)·Ψ(n_(c)·θ)+F_(c)(n_(c)·θ+Δ)·Ψ(n_(c)·θ+Δ) to be constantas a function of θ.
 7. The method of claim 1, wherein the cam assemblypresents to the first linear actuator a first cam surface having a firstprofile and presents to the second linear actuator a second cam surfacehaving a second profile, wherein the first profile, the second profile,the first drive signal and the second drive signal are selected toresult in the total torque being substantially ripple free throughoutthe complete rotation of the rotational shaft.
 8. The method of claim 7,wherein the first profile is described by n cycles of a trigonometricfunction, wherein n is an even integer.
 9. The method of claim 8,wherein the second profile is described by n cycles of saidtrigonometric function.
 10. The method of claim 9, wherein n equals 4.11. The method of claim 9, wherein said trigonometric function is a sinefunction.